Question

Aratati ca:
$1 + sinx = 2 \cos^{2} ( \frac{\pi}{4} - \frac{x}{2} )$

DAU COROANA!!​

Answer 1

Răspuns:

Explicație pas cu pas:

$1 + sinx = 2 \cos^{2} ( \frac{\pi}{4} - \frac{x}{2} )\\cos( \frac{\pi}{4} - \frac{x}{2} )=cos\frac{\pi}{4}\cdot cos \frac{x}{2}+sin \frac{x}{2}\cdot sin \frac{\pi}{4} =\frac{\sqrt{2} }{2} \cdot (cos \frac{x}{2}+sin \frac{x}{2})$

$sinx=2sin\frac{x}{2} \cdot cos\frac{x}{2}$

$cos^{2}(\frac{\pi}{4}-\frac{x}{2})=\frac{1}{2} \cdot (cos^{2} \frac{x}{2} +sin^{2} \frac{x}{2} +2sin\frac{x}{2} cos\frac{x}{2} )=\frac{1}{2} \cdot (1+sinx)$

Deci

$2cos^{2}(\frac{\pi}{4}-\frac{x}{2})=2\cdot\frac{1}{2} \cdot (1+sinx)=1+sinx$